Model averaging procedure for varying-coefficient partially linear models with missing responses

This paper is concerned with model averaging procedure for varying-coefficient partially linear models with missing responses. The profile least-squares estimation process and inverse probability weighted method are employed to estimate regression coefficients of the partially restricted models, in which the propensity score is estimated by the covariate balancing propensity score method. The estimators of the linear parameters are shown to be asymptotically normal. Then we develop the focused information criterion, formulate the frequentist model averaging estimators and construct the corresponding confidence intervals. Some simulation studies are conducted to examine the finite sample performance of the proposed methods. We find that the covariate balancing propensity score improves the performance of the inverse probability weighted estimator. We also demonstrate the superiority of the proposed model averaging estimators over those of existing strategies in terms of mean squared error and coverage probability. Finally, our approach is further applied to a real data example.     

Semiparametric Proportional Mean Residual Life Model With Censoring Indicators Missing at Random

For right-censored survival data, the information that whether the observed time is survival or censoring time is frequently lost. This is the case for the competing risk data. In this article, we consider statistical inference for the right-censored survival data with censoring indicators missing at random under the proportional mean residual life model. Simple and augmented inverse probability weighted estimating equation approaches are developed, in which the nonmissingness probability and some unknown conditional expectations are estimated by the kernel smoothing technique. The asymptotic properties of all the proposed estimators are established, while extensive simulation studies demonstrate that our proposed methods perform well under the moderate sample size. At last, the proposed method is applied to a data set from a stage II breast cancer trial.     

Efficient estimation of the PDF and the CDF of the inverse Rayleigh distribution

In this paper, we consider the estimation of the probability density function and the cumulative distribution function of the inverse Rayleigh distribution. In this regard, the following estimators are considered: uniformly minimum variance unbiased estimator, maximum likelihood (ML) estimator, percentile estimator, least squares estimator and weighted least squares estimator. To do so, analytical expressions are derived for the mean integrated squared error. As the result of simulation studies and real data applications indicate, when the sample size is not very small the ML estimator performs better than the others.     

Improved estimation methods for unrelated question randomized response techniques

The randomized response technique (RRT) is an important tool, commonly used to avoid biased answers in survey on sensitive issues by preserving the respondents’ privacy. In this paper, we introduce a data collection method for survey on sensitive issues combining both the unrelated-question RRT and the direct question design. The direct questioning method is utilized to obtain responses to a non sensitive question that is related to the innocuous question from the unrelated-question RRT. These responses serve as additional information that can be used to improve the estimation of the prevalence of the sensitive behavior. Furthermore, we propose two new methods for the estimation of the proportion of respondents possessing the sensitive attribute under a missing data setup. More specifically, we develop the weighted estimator and the weighted conditional likelihood estimator. The performances of our estimators are studied numerically and compared with that of an existing one. Both proposed estimators are more efficient than the Greenberg's estimator. We illustrate our methods using real data from a survey study on illegal use of cable TV service in Taiwan.     

Statistical inference for varying-coefficient partially linear errors-in-variables models with missing data

Abstract

The purpose of this paper is twofold. First, we investigate estimations in varying-coefficient partially linear errors-in-variables models with covariates missing at random. However, the estimators are often biased due to the existence of measurement errors, the bias-corrected profile least-squares estimator and local liner estimators for unknown parametric and coefficient functions are obtained based on inverse probability weighted method. The asymptotic properties of the proposed estimators both for the parameter and nonparametric parts are established. Second, we study asymptotic distributions of an empirical log-likelihood ratio statistic and maximum empirical likelihood estimator for the unknown parameter. Based on this, more accurate confidence regions of the unknown parameter can be constructed. The methods are examined through simulation studies and illustrated by a real data analysis.     

Estimation of average treatment effects based on parametric propensity score model

In this paper, the estimation of average treatment effects is examined given that the propensity score is of a parametric form with some unknown parameters. Under the assumption that the treatment is ignorable given some observed characteristics, the MLEs for those unknown parameters in the probability assignment model have been achieved firstly and then three estimators have been defined by the inverse probability weighted, regression and imputation methods, respectively. All the estimators are shown asymptotically normal and more importantly, the substantial efficiency gains of the first two estimates have been obtained theoretically compared with the existing estimators in Hahn (1998) and Hirano et al. (2003), i.e., the inverse weighted probability estimator and the regression estimator have smaller asymptotic variances. Our simulation analysis verifies the theoretical results in terms of biases, SEs and MSEs.     

Estimation in a linear regression model with stochastic linear restrictions: a new two-parameter-weighted mixed estimator

The present paper considers the weighted mixed regression estimation of the coefficient vector in a linear regression model with stochastic linear restrictions binding the regression coefficients. We introduce a new two-parameter-weighted mixed estimator (TPWME) by unifying the weighted mixed estimator of Schaffrin and Toutenburg [1] and the two-parameter estimator (TPE) of Özkale and Kaç?ranlar [2]. This new estimator is a general estimator which includes the weighted mixed estimator, the TPE and the restricted two-parameter estimator (RTPE) proposed by Özkale and Kaç?ranlar [2] as special cases. Furthermore, we compare the TPWME with the weighted mixed estimator and the TPE with respect to the matrix mean square error criterion. A numerical example and a Monte Carlo simulation experiment are presented by using different estimators of the biasing parameters to illustrate some of the theoretical results.     

CBPS-based estimation for linear models with responses missing at random

In this article, based on the covariate balancing propensity score (CBPS), estimators for the regression coefficients and the population mean are obtained, when the responses of linear models are missing at random. It is proved that the proposed estimators are asymptotically normal. In simulation studies and real example, the proposed estimators show improved performance relative to usual augmented inverse probability weighted estimators.     

Probability density estimation with data missing at random when covariables are present

This paper addresses the problem of the probability density estimation in the presence of covariates when data are missing at random (MAR). The inverse probability weighted method is used to define a nonparametric and a semiparametric weighted probability density estimators. A regression calibration technique is also used to define an imputed estimator. It is shown that all the estimators are asymptotically normal with the same asymptotic variance as that of the inverse probability weighted estimator with known selection probability function and weights. Also, we establish the mean squared error (MSE) bounds and obtain the MSE convergence rates. A simulation is carried out to assess the proposed estimators in terms of the bias and standard error.     

Reweighted estimators for additive hazard model with censoring indicators missing at random

Survival data with missing censoring indicators are frequently encountered in biomedical studies. In this paper, we consider statistical inference for this type of data under the additive hazard model. Reweighting methods based on simple and augmented inverse probability are proposed. The asymptotic properties of the proposed estimators are established. Furthermore, we provide a numerical technique for checking adequacy of the fitted model with missing censoring indicators. Our simulation results show that the proposed estimators outperform the simple and augmented inverse probability weighted estimators without reweighting. The proposed methods are illustrated by analyzing a dataset from a breast cancer study.     

A polar coordinate transformation for estimating bivariate survival functions with randomly censored and truncated data

This paper proposes a new estimator for bivariate distribution functions under random truncation and random censoring. The new method is based on a polar coordinate transformation, which enables us to transform a bivariate survival function to a univariate survival function. A consistent estimator for the transformed univariate function is proposed. Then the univariate estimator is transformed back to a bivariate estimator. The estimator converges weakly to a zero-mean Gaussian process with an easily estimated covariance function. Consistent truncation probability estimate is also provided. Numerical studies show that the distribution estimator and truncation probability estimator perform remarkably well.     

Penalized inverse probability weighted estimators for weighted rank regression with missing covariates

Abstract

In this article, we study the variable selection and estimation for linear regression models with missing covariates. The proposed estimation method is almost as efficient as the popular least-squares-based estimation method for normal random errors and empirically shown to be much more efficient and robust with respect to heavy tailed errors or outliers in the responses and covariates. To achieve sparsity, a variable selection procedure based on SCAD is proposed to conduct estimation and variable selection simultaneously. The procedure is shown to possess the oracle property. To deal with the covariates missing, we consider the inverse probability weighted estimators for the linear model when the selection probability is known or unknown. It is shown that the estimator by using estimated selection probability has a smaller asymptotic variance than that with true selection probability, thus is more efficient. Therefore, the important Horvitz-Thompson property is verified for penalized rank estimator with the covariates missing in the linear model. Some numerical examples are provided to demonstrate the performance of the estimators.     

Estimation of Stratified Mark‐Specific Proportional Hazards Models with Missing Marks

Abstract. An objective of randomized placebo‐controlled preventive HIV vaccine efficacy trials is to assess the relationship between the vaccine effect to prevent infection and the genetic distance of the exposing HIV to the HIV strain represented in the vaccine construct. Motivated by this objective, recently a mark‐specific proportional hazards (PH) model with a continuum of competing risks has been studied, where the genetic distance of the transmitting strain is the continuous ‘mark’ defined and observable only in failures. A high percentage of genetic marks of interest may be missing for a variety of reasons, predominantly because rapid evolution of HIV sequences after transmission before a blood sample is drawn from which HIV sequences are measured. This research investigates the stratified mark‐specific PH model with missing marks where the baseline functions may vary with strata. We develop two consistent estimation approaches, the first based on the inverse probability weighted complete‐case (IPW) technique, and the second based on augmenting the IPW estimator by incorporating auxiliary information predictive of the mark. We investigate the asymptotic properties and finite‐sample performance of the two estimators, and show that the augmented IPW estimator, which satisfies a double robustness property, is more efficient.     

Empirical likelihood weighted composite quantile regression with partially missing covariates

This paper develops a novel weighted composite quantile regression (CQR) method for estimation of a linear model when some covariates are missing at random and the probability for missingness mechanism can be modelled parametrically. By incorporating the unbiased estimating equations of incomplete data into empirical likelihood (EL), we obtain the EL-based weights, and then re-adjust the inverse probability weighted CQR for estimating the vector of regression coefficients. Theoretical results show that the proposed method can achieve semiparametric efficiency if the selection probability function is correctly specified, therefore the EL weighted CQR is more efficient than the inverse probability weighted CQR. Besides, our algorithm is computationally simple and easy to implement. Simulation studies are conducted to examine the finite sample performance of the proposed procedures. Finally, we apply the new method to analyse the US news College data.     

On equalities between BLUES,WLSEs, and SLSEs

Necessary and sufficient conditions for equalities between the best linear unbiased estimator, the weighted least-squares estimator, and the simple least-squares estimator of the expectation vector in a general Gauss-Markoff model are given in some alternative formulations. The main result states, somewhat surprisingly, that the weighted least-squares estimator cannot be identical with the simple least-squares estimator unless they both coincide with the best linear unbiased estimator.     

Empirical likelihood confidence bands for distribution functions with missing responses

Distribution function estimation plays a significant role of foundation in statistics since the population distribution is always involved in statistical inference and is usually unknown. In this paper, we consider the estimation of the distribution function of a response variable Y with missing responses in the regression problems. It is proved that the augmented inverse probability weighted estimator converges weakly to a zero mean Gaussian process. A augmented inverse probability weighted empirical log-likelihood function is also defined. It is shown that the empirical log-likelihood converges weakly to the square of a Gaussian process with mean zero and variance one. We apply these results to the construction of Gaussian process approximation based confidence bands and empirical likelihood based confidence bands of the distribution function of Y. A simulation is conducted to evaluate the confidence bands.     

Focused information criterion and model averaging based on weighted composite quantile regression

We study the focused information criterion and frequentist model averaging and their application to post‐model‐selection inference for weighted composite quantile regression (WCQR) in the context of the additive partial linear models. With the non‐parametric functions approximated by polynomial splines, we show that, under certain conditions, the asymptotic distribution of the frequentist model averaging WCQR‐estimator of a focused parameter is a non‐linear mixture of normal distributions. This asymptotic distribution is used to construct confidence intervals that achieve the nominal coverage probability. With properly chosen weights, the focused information criterion based WCQR estimators are not only robust to outliers and non‐normal residuals but also can achieve efficiency close to the maximum likelihood estimator, without assuming the true error distribution. Simulation studies and a real data analysis are used to illustrate the effectiveness of the proposed procedure.     

Weighted empirical likelihood for quantile regression with non ignorable missing covariates

In this paper, we propose an empirical likelihood-based weighted estimator of regression parameter in quantile regression model with non ignorable missing covariates. The proposed estimator is computationally simple and achieves semiparametric efficiency if the probability of missingness on the fully observed variables is correctly specified. The efficiency gain of the proposed estimator over the complete-case-analysis estimator is quantified theoretically and illustrated via simulation and a real data application.     

A Model Validation Procedure when Covariate Data are Missing at Random

Abstract. In the presence of missing covariates, standard model validation procedures may result in misleading conclusions. By building generalized score statistics on augmented inverse probability weighted complete‐case estimating equations, we develop a new model validation procedure to assess the adequacy of a prescribed analysis model when covariate data are missing at random. The asymptotic distribution and local alternative efficiency for the test are investigated. Under certain conditions, our approach provides not only valid but also asymptotically optimal results. A simulation study for both linear and logistic regression illustrates the applicability and finite sample performance of the methodology. Our method is also employed to analyse a coronary artery disease diagnostic dataset.     

Analysis of two-phase sampling data with semiparametric additive hazards models

Under the case-cohort design introduced by Prentice (Biometrica 73:1–11, 1986), the covariate histories are ascertained only for the subjects who experience the event of interest (i.e., the cases) during the follow-up period and for a relatively small random sample from the original cohort (i.e., the subcohort). The case-cohort design has been widely used in clinical and epidemiological studies to assess the effects of covariates on failure times. Most statistical methods developed for the case-cohort design use the proportional hazards model, and few methods allow for time-varying regression coefficients. In addition, most methods disregard data from subjects outside of the subcohort, which can result in inefficient inference. Addressing these issues, this paper proposes an estimation procedure for the semiparametric additive hazards model with case-cohort/two-phase sampling data, allowing the covariates of interest to be missing for cases as well as for non-cases. A more flexible form of the additive model is considered that allows the effects of some covariates to be time varying while specifying the effects of others to be constant. An augmented inverse probability weighted estimation procedure is proposed. The proposed method allows utilizing the auxiliary information that correlates with the phase-two covariates to improve efficiency. The asymptotic properties of the proposed estimators are established. An extensive simulation study shows that the augmented inverse probability weighted estimation is more efficient than the widely adopted inverse probability weighted complete-case estimation method. The method is applied to analyze data from a preventive HIV vaccine efficacy trial.